# Properties

 Label 4008.359 Modulus $4008$ Conductor $2004$ Order $166$ Real no Primitive no Minimal no Parity even

# Related objects

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(4008)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([83,0,83,2]))

pari: [g,chi] = znchar(Mod(359,4008))

## Basic properties

 Modulus: $$4008$$ Conductor: $$2004$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$166$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{2004}(359,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 4008.bc

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(3007,2005,1337,673)$$ → $$(-1,1,-1,e\left(\frac{1}{83}\right))$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$1$$ $$1$$ $$e\left(\frac{85}{166}\right)$$ $$e\left(\frac{153}{166}\right)$$ $$e\left(\frac{28}{83}\right)$$ $$e\left(\frac{20}{83}\right)$$ $$e\left(\frac{23}{166}\right)$$ $$e\left(\frac{33}{166}\right)$$ $$e\left(\frac{16}{83}\right)$$ $$e\left(\frac{2}{83}\right)$$ $$e\left(\frac{51}{166}\right)$$ $$e\left(\frac{97}{166}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{83})$ Fixed field: Number field defined by a degree 166 polynomial